AW.: RE: RE: RE: Harmonic entropy

In einer Nachricht vom 11/17/99 1:31:00 AM (MEZ) Mitteleurop�ische
Zeitschreibt PErlich@Acadian-Asset.com:

<<
Thanks, Daniel. There are many differences between Barlow's idea and
harmonic entropy. Harmonic entropy makes no reference to a scale, referring
instead to a simultaneous dyad presented with no musical context.
>>

I understand that. But unless considered in a particular context, I can't
ascribe much importance to a single interval, and Barlough seems to be the
first to propose some criteria for evaluating such a context.

<<>>

Your argument makes a strange circle here. Any given limit is an assumption
of greater importance for simpler ratios in that the limit is (a) a limit and
(b) the distribution of prime factors in any given octave will be biased
towards the lower factors. (Two stupid observations on my part). While the
entropy idea may have no such assumption in itself, by using the Farey
series for your calculations you are are stuck with this bias. I don't happen
to disagree with the results, but you won't get a significantly different
answer by taking a set similar to that chosen by Barlow.

<<< So even if _a priori_ we can recognize
ratios of arbitrary complexity (which is what the harmonic entropy model
assumes), the imprecision of our hearing mechanism will spread any stimulus
over a finite range in interval space. If the true interval is near a simple
ratio, there won't be much confusion as to which ratio is heard; otherwise,
many ratios will be included within the "smear" and the listener is
confused.
<<<

This "confusion" doesn't conform to any real listening experience I've ever
had. In experimental settings -- my work with all those minor thirds last
year, for example -- the length of the listening period was everything in
sorting out a series of intervals. My sensation in analysing an interval
over time is not that I am sorting through a smear of possible identities but
rather that a single identity is gradually speaking, and that identity is not
isolated from beating or difference tones. In real musical performances,
where the smear is often really there (portamenti, vibrato) such rarified
listening cannot take place, context is everything, and I find myself in need
of something like Barlow's second condition

<< Harmonic entropy models dissonance with a mathematical measure of
this confusion, namely the entropy of the set of probabilities associated
with the proportion of the "smear" that is assigned to the various ratios.
Entropy is the natural measure for disorder and also for informational
complexity. >>

But similar to Tenney's use of entropy as a measure of variation in musical
form, or the various applications of cybernetic theory in analysis used by
Franz-Jochen Herfert, I am still left with the feeling that a power tool is
being used to tackle a task that can best be done manually

<< Now it may be that simple ratios _are_ more important even apart from the
"smearing", in which case harmonic entropy gives too much importance to
complex ratios. Using a standard deviation of 0.6% is also quite ideal; even
the best listener in Goldstein's experiment only acheived this accuracy in
at some optimal frequency (was it 2000Hz) and did worse than 2% in other
musically relevant areas. So the results I presented were absolutely the
best-case scenario for perceiving complex ratios, aside from
combination-tone effects. I think this is an important exercise before
plunging into the endless array of ratios that are our disposal for
constructing JI systems or evaluating tempered systems.
>>

I am certain that Goldstein was working with very limited sample periods and
a realistic evaluation of musical listening is going to have to take this
into account, as well as work with timbres where beating becomes unavoidable.